The Emergence of Mathematical Structuralism

The project will investigate the historical roots of mathematical structuralism, one of the most prominent positions in contemporary philosophy of mathematics. The positions core idea is that mathematical theories describe abstract structures: Peano arithmetic describes the natural number structure, analysis the real number structure, geometry the structure of (Euclidian) space, and so on. Precise elaborations, or formal explications, of that idea have been provided by a number of current philosophers of mathematics, resulting in a number of different versions of structuralism (e.g. Shapiro 1997, Resnik 1997, Parsons 2009). The principal objective in this project is to provide a first systematic study of the mathematical and philosophical origins of this philosophical position. Specifically, the focus will be set on two historical developments and their consequences for the present debate: the first concerns several conceptual changes in geometry between 1860 and 1900 that eventually led to a more general ``structuralist turn`` in mathematics. A specific focus will be put here on the gradual implementation of model-theoretic techniques in geometrical reasoning as well as on the classification of geometrical theories in terms of invariants as specified in Kleins Erlangen program. The second development concerns the early philosophical reflection on these mathematical transformations between 1910 and 1940. This includes different attempts by thinkers such as Rudolf Carnap, Edmund Husserl and Ernst Cassirer to spell out the philosophical implications of the new structuralist methodologies at work in modern geometry. These early contributions to mathematical structuralism have so far been largely ignored in the modern debate. The project will provide a first systematic and comparative study of them as well as of their immediate background in nineteenth-century geometry. Specifically, it will investigate the following questions: How did Husserl, Cassirer, and Carnap characterize the notion of abstract mathematical structures, or the structural content of mathematical theories? Are there points of contact between their respective positions, as well as between these philosophical positions and earlier developments within mathematics? Finally, how are these early contributions to structuralism related to the discussions in current philosophy of mathematics, in particular on the proper structuralist ontology and epistemology of mathematical objects?

The project presents a first contribution to the investigation of the historical emergence of mathematical structuralism, one of the most prominent positions in contemporary philosophy of mathematics. The positions core idea is that mathematical theories describe abstract structures: Peano arithmetic describes the natural number structure, analysis the real number structure, geometry the structure of (Euclidian) space, and so on. Precise elaborations, or formal explications, of that idea have been provided by a number of current philosophers of mathematics, resulting in a number of different versions of structuralism. The principal objective in this project was to give a first systematic study of the mathematical and philosophical origins of this philosophical position. Specifically, the focus was set on two historical developments and their consequences for the present debate: the first concerns several conceptual changes in geometry between 1860 and 1900 that eventually led to a general structural turn in mathematics. A specific focus was put here on the gradual implementation of model-theoretic techniques in axiomatic reasoning as well as on the classification of geometrical theories in terms of group-theoretic. The second development investigated in the course of the project concerns the early philosophical reflection on these mathematical transformations between 1910 and 1940. This includes different attempts by thinkers such as Rudolf Carnap and Ernst Cassirer to spell out the philosophical implications of the new structuralist methodologies at work in modern geometry. The project presents a first attempt of a systematic and comparative study of these philosophical developments and their immediate background in nineteenth-century geometry. Specifically, the following questions have been addressed here: How did Cassirer and Carnap characterize the notion of abstract mathematical structures, or the structural content of mathematical theories? Are there points of contact between their respective positions, as well as between these philosophical positions and earlier developments within mathematics? The project also analyzed how are these early contributions to structuralism can be related to discussions in current philosophy of mathematics, in particular on the proper structuralist ontology and epistemology of mathematical objects as well as on the proper understanding of notions such as structure abstraction and structural properties.